Gödel's incompleteness theorems
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The Age of Reason sought to establish axiomatic philosophy and absolutism as foundations for knowledge and stability. Epistemology, in the writings of Michel de Montaigne and René Descartes, was based on extreme skepticism and inquiry into the nature of "knowledge." The goal of a philosophy based on self-evident axioms reached its height with Baruch (Benedictus de) Spinoza's Ethics, which expounded a pantheistic view of the universe where God and Nature were one. This idea then became central to the Enlightenment from Newton through to Jefferson.
Issues
- "We're tired of trees. We should stop believing in trees, roots, and radicles. They've made us suffer too much. All of arborescent culture is founded on them, from biology to linguistics" --A Thousand Plateaus: Capitalism and Schizophrenia --Deleuze & Guattari
Not every consistent body of propositions can be captured by a describable collection of axioms. Call a collection of axioms recursive if a computer program can recognize whether a given proposition in the language is an axiom. Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. If the computer cannot recognize the axioms, the computer also will not be able to recognize whether a proof is valid. The result is that one will not know which propositions are theorems and the axiomatic method breaks down. An example of such a body of propositions is the theory of the natural numbers. The Peano Axioms thus only partially axiomatize this theory.
See also
